Calculus Examples

$\frac{1}{1 + \cos (x)}$

Step 1

Write $\frac{1}{1 + \cos (x)}$ as a function.

$f (x) = \frac{1}{1 + \cos (x)}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{1}{1 + \cos (x)} d x$

Step 4

Use the double-angle identity to transform $\cos (x)$ to $\cos^{2} (\frac{x}{2}) - \sin^{2} (\frac{x}{2})$ .

$\int \frac{1}{1 + \cos^{2} (\frac{x}{2}) - \sin^{2} (\frac{x}{2})} d x$

Step 5

Use the pythagorean identity to transform $1$ to $\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2})$ .

$\int \frac{1}{\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) - \sin^{2} (\frac{x}{2})} d x$

Step 6

Simplify.

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Step 6.1

Subtract ${\sin (\frac{x}{2})}^{2}$ from ${\sin (\frac{x}{2})}^{2}$ .

$\int \frac{1}{{\cos (\frac{x}{2})}^{2} + {\cos (\frac{x}{2})}^{2} + 0} d x$

Step 6.2

Add ${\cos (\frac{x}{2})}^{2}$ and ${\cos (\frac{x}{2})}^{2}$ .

$\int \frac{1}{2 {\cos (\frac{x}{2})}^{2} + 0} d x$

Step 6.3

Add $2 {\cos (\frac{x}{2})}^{2}$ and $0$ .

$\int \frac{1}{2 \cos^{2} (\frac{x}{2})} d x$

Step 7

Multiply the argument by $\frac{\sec^{2} (\frac{x}{2})}{\sec^{2} (\frac{x}{2})}$

$\int \frac{1}{2 \cos^{2} (\frac{x}{2})} \cdot \frac{\sec^{2} (\frac{x}{2})}{\sec^{2} (\frac{x}{2})} d x$

Step 8

Combine.

$\int \frac{1 \sec^{2} (\frac{x}{2})}{2 \cos^{2} (\frac{x}{2}) \sec^{2} (\frac{x}{2})} d x$

Step 9

Multiply ${\sec (\frac{x}{2})}^{2}$ by $1$ .

$\int \frac{\sec^{2} (\frac{x}{2})}{2 \cos^{2} (\frac{x}{2}) \sec^{2} (\frac{x}{2})} d x$

Step 10

Rewrite $\sec (\frac{x}{2})$ in terms of sines and cosines.

$\int \frac{\sec^{2} (\frac{x}{2})}{2 \cos^{2} (\frac{x}{2}) {(\frac{1}{\cos (\frac{x}{2})})}^{2}} d x$

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

Apply the product rule to $\frac{1}{\cos (\frac{x}{2})}$ .

$\int \frac{\sec^{2} (\frac{x}{2})}{2 \cos^{2} (\frac{x}{2}) \frac{1^{2}}{\cos^{2} (\frac{x}{2})}} d x$

Step 11.2

Cancel the common factor of $\cos^{2} (\frac{x}{2})$ .

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Step 11.2.1

Factor $\cos^{2} (\frac{x}{2})$ out of $2 \cos^{2} (\frac{x}{2})$ .

$\int \frac{\sec^{2} (\frac{x}{2})}{\cos^{2} (\frac{x}{2}) \cdot 2 \frac{1^{2}}{\cos^{2} (\frac{x}{2})}} d x$

Step 11.2.2

Cancel the common factor.

$\int \frac{\sec^{2} (\frac{x}{2})}{\cos^{2} (\frac{x}{2}) \cdot 2 \frac{1^{2}}{\cos^{2} (\frac{x}{2})}} d x$

Step 11.2.3

Rewrite the expression.

$\int \frac{\sec^{2} (\frac{x}{2})}{2 \cdot 1^{2}} d x$

Step 11.3

Simplify the expression.

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Step 11.3.1

One to any power is one.

$\int \frac{\sec^{2} (\frac{x}{2})}{2 \cdot 1} d x$

Step 11.3.2

Multiply $2$ by $1$ .

$\int \frac{\sec^{2} (\frac{x}{2})}{2} d x$

Step 12

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \sec^{2} (\frac{x}{2}) d x$

Step 13

Let $u = \frac{x}{2}$ . Then $d u = \frac{1}{2} d x$ , so $2 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 13.1

Let $u = \frac{x}{2}$ . Find $\frac{d u}{d x}$ .

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Step 13.1.1

Differentiate $\frac{x}{2}$ .

$\frac{d}{d x} [\frac{x}{2}]$

Step 13.1.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$\frac{1}{2} \frac{d}{d x} [x]$

Step 13.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{2} \cdot 1$

Step 13.1.4

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2}$

Step 13.2

Rewrite the problem using $u$ and $d u$ .

$\frac{1}{2} \int \sec^{2} (u) \frac{1}{\frac{1}{2}} d u$

Step 14

Simplify.

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Step 14.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\frac{1}{2} \int \sec^{2} (u) (1 \cdot 2) d u$

Step 14.2

Multiply $2$ by $1$ .

$\frac{1}{2} \int \sec^{2} (u) \cdot 2 d u$

Step 14.3

Move $2$ to the left of $\sec^{2} (u)$ .

$\frac{1}{2} \int 2 \sec^{2} (u) d u$

Step 15

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$\frac{1}{2} (2 \int \sec^{2} (u) d u)$

Step 16

Simplify.

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Step 16.1

Combine $2$ and $\frac{1}{2}$ .

$\frac{2}{2} \int \sec^{2} (u) d u$

Step 16.2

Cancel the common factor of $2$ .

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Step 16.2.1

Cancel the common factor.

$\frac{2}{2} \int \sec^{2} (u) d u$

Step 16.2.2

Rewrite the expression.

$1 \int \sec^{2} (u) d u$

Step 16.3

Multiply $\int \sec^{2} (u) d u$ by $1$ .

$\int \sec^{2} (u) d u$

Step 17

Since the derivative of $\tan (u)$ is $\sec^{2} (u)$ , the integral of $\sec^{2} (u)$ is $\tan (u)$ .

$\tan (u) + C$

Step 18

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$\tan (\frac{x}{2}) + C$

Step 19

Reorder terms.

$\tan (\frac{1}{2} x) + C$

Step 20

The answer is the antiderivative of the function $f (x) = \frac{1}{1 + \cos (x)}$ .

$F (x) =$ $\tan (\frac{1}{2} x) + C$